Cross model, typical parameters

Figure 12 The Cross model with m = 1 and typical values ofthe parameters, showing the apparent yield stress region and upper and lower Newtonian regions.

Model Networks. Constmction of model networks allows development of quantitative stmcture property relationships and provide the abiUty to test the accuracy of the theories of mbber elasticity (251—254). By definition, model networks have controlled molecular weight between cross-links, controlled cross-link functionahty, and controlled molecular weight distribution of cross-linked chains. Sihcones cross-linked by either condensation or addition reactions are ideally suited for these studies because all of the above parameters can be controlled. A typical condensation-cure model network consists of an a, CO-polydimethylsiloxanediol, tetraethoxysilane (or alkyltrimethoxysilane), and a tin-cure catalyst (255). A typical addition-cure model is composed of a, ffl-vinylpolydimethylsiloxane, tetrakis(dimethylsiloxy)silane, and a platinum-cure catalyst (256—258). 

Because estimation of model parameters, b0, bx,. .., bm uses m + 1 degrees of freedom, the remaining n-m- 1 degrees of freedom are used to estimate RMSEC. If the intercept b, is omitted from the calibration model, then the number of degrees of freedom for RMSEC is n-m. If the data has been mean-centered, the degrees of freedom remain n-m-1. Typically, RMSEC provides overly optimistic estimates of a calibration model s predictive ability for samples measured in the future. This is because a portion of the noise in the standards is inadvertently modeled by the estimated parameters. A better estimate of the calibration model s predictive ability may be obtained by the method of cross-validation with the calibration samples or from a separate set of validation samples. 

NIR spectroscopy became much more useful when the principle of multiple-wavelength spectroscopy was combined with the deconvolution methods of factor and principal component analysis. In typical applications, partial least squares regression is used to model the relation between composition and the NIR spectra of an appropriately chosen series of calibration samples, and an optimal model is ultimately chosen by a procedure of cross-testing. The performance of the optimal model is then evaluated using the normal analytical performance parameters of accuracy, precision, and linearity. Since its inception, NIR spectroscopy has been viewed primarily as a technique of quantitative analysis and has found major use in the determination of water in many pharmaceutical materials. 

Entrainment Dryers In design mode, the required gas flow rate can be obtained from a heat and mass balance. For pneumatic conveying dryers, duct cross-sectional area and diameter are found from the scoping design calculation (if required gas velocity is unknown, a typical value is 20 m/s). Duct length can be estimated by an incremental model, but some parameters are hard to obtain and conditions change rapidly near the feed point, so the model is most effective for scaling up from pilot-plant data see Kemp and Oakley (2002). Spray... 

It is not necessary to model individual pipes, if the cross sections are represented by a typical diameter. In any case the primary parameter of interest is the volume (Eq. 6.119) or the dead time (Eq. 6.120) of the unit. 

The design allows checking the reliabihty of the model fitted to the data, typically by statistical tests about the model parameters and the model adequacy (lack-of-fit), and by cross validation. 

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